This invention relates to communications systems and in particular to a spatial diversity receiving system for receiving a digitally modulated signal.
There are several digital modulation techniques, including BPSK, QAM, (sometimes referred to as QASK), QPSK, which can be demodulated by a delay and multiply operation known as differential detection. In point-to-point and point-to-multi-point communications systems, particularly mobile radio systems, it is known that the medium""s channel properties vary over time. These channel variations cause the receiver signal to fade in and out or be subject to intermittent outages. Examples of media or communications channels exhibiting fading properties include electromagnetic signals propagating through the Earth""s atmosphere, or undersea acoustic signals.
Signal fading in communications channels is a natural phenomenon that limits the range of separation between the transmitter and receiver. Many very practical and useful channels exhibit fading. The fading in these channels is usually caused by the receiver being linked to the transmitter by more than one propagation path, the lengths of which vary with time. Such channels usually exhibit filing properties terminated Rayleigh fadfing or Rician fading. Fading can also be caused by time varying absorption. For example, the absorption properties of an atmospheric radio channel depend on the moisture content of the atmosphere, which changes over time.
The reception over a channel subject to fading, can be improved by incorporating diversity to reduce the fading and intermittent channel effects that interfere with communications over the channel. There are several diversity options including multiple transmissions using the same communication frequency with each transmission separated in time from the other, which may be termed time diversity; multiple carrier frequencies requiring several different communication frequencies, which may be termed frequency diversity; and multiple antennas or receiving elements with each antenna or receiving element physically separate from the other, which may be termed spatial diversity; etc.
It is well known that the reception of a signal that propagates through a fading or intermittent channel can be improved through the use of spatial diversity. Spatial diversity systems utilize multiple, physically separated, receiving elements in a way that mitigates the fading or intermittent outages experienced by each of the receiving elements. A strong persistent signal, with a much smaller degree of fading, is obtained by properly processing the signals from each of the receiving elements and then properly combining these signals. The strength and persistence of the combined signal depends on the number of receiving elements and the techniques used to process and combine the signals. There is a variety of signal processing and combining techniques that can be used.
One approach to reducing signal fading is to use some form of diversity to receive multiple signals at the receiver and then to combine these signals in a constructive way. A diversity system could involve frequency diversity which is implemented by transmitting identical information on two or more separate carriers that are separated enough in frequency for the fading of each carrier to be uncorrelated with the others. It could involve sending the identical information on the same carrier but at two or more different times with enough separation in time for the fading at each time to be uncorrelated (time diversity). It could also involve sending the information only once, on one carrier and receiving it on two or more separate antennas physically separated with enough distance between them for the fading at each antenna in the receiver to be uncorrelated with the others (spatial diversity).
In all diversity systems the multiple signals have to be combined, which usually involves processing or conditioning each of the multiple signals and then summing or selecting these processed signals. There are different methods for combining the signals, each offering different trade offs between performance and implementation complexity.
Each of the three types of diversity has its advantages and disadvantages. However all three types require a diversity combiner. Spatial diversity has the strong advantage of requiring minimal bandwidth to transmit the information. It has the disadvantage in radio frequency communication of requiring more high frequency circuitry than the others, in particular multiple antennas and associated radio frequency (RF) electronics. Spatial diversity is particularly attractive when the carrier frequency is sufficiently high for the antennas to be implemented as printed circuits. At these frequencies the cost of implementing spatial diversity is the cost of the electronics associated with each antenna.
The theory of spatial diversity and its ability to mitigate fading is well known. To reach the theoretical performance limit requires knowledge of the amplitude and phase of the carrier, which changes with time and can only be estimated. The challenge in achieving near optimal performance is in the implementation of an amplitude and phase estimator and the implementation of the phase and amplitude corrector. There are many techniques for estimating and correcting amplitude and phase and for combining the corrected signals. The various techniques have different degrees of compromise between performance and ease of implementation. Examples of prior art that fall into this category include:
U.S. Pat. No. 4,386,435 to Ulmer providing a space diversity receiver includes an IF band combiner amplifier to sum the IF signals where one signal has a phase corrector which adjusts the phase in response to received signal characteristics. A similar approach is also used in U.S. Pat. Nos. 4,326,294 and 4,710,975 both to Okamoto et al.
U.S. Pat. No. 5,530,925 to Garner combines signals from two physically diverse antennas after down conversion to an intermediate frequency. U.S. Pat. No. 4,498,885 to Namiki combines two spatially diverse signals relying on controlled phase shifting of one of the signals to cancel the effects of an interference wave.
In U.S. Pat. No. 5,203,025 to Anvari et al. the relative phase and amplitude of the IF stage of a spatially diverse receiver are used to determine a signal combining strategy of either direct summation, or inversion then summation, for recovering the modulating signal. A related approach is employed by Karabinis in U.S. Pat. No. 4,373,210 which selects from two spatially diverse signals based on relative signal to interference ratios. The receiver in U.S. Pat. No. 4,079,318 to Kinoshita combines signals from spatially diverse antennas relying on phase control at the intermediate frequency stage.
There are several techniques for combining the signals from multiple antennas to mitigate the effects of fading. These techniques can be logically divided into two categories: post-detection combining and pre-detection combining. The post-detection combiners essentially require an entire receiver for each antenna but the decision and control circuits remain common. While these circuits can achieve near optimum performance, they are expensive solutions. Pre-detection schemes can also be very complicated and expensive. To achieve near optimum performance requires circuits that estimate the phase of the carrier received at each antenna as well as voltage controlled phase shifter circuits to perform in phase alignment or correction. The purpose of these circuits is to align the phase of the carriers received on each antenna. The implementation cost of such systems is quite high.
A simpler method that yields suboptimum performance can be described as a frequency stacked IF combiner. In this method the carriers from each antenna are translated to separate IF frequencies, with the separation between IF carriers being multiples of the bit rate. The multiplicity of frequency stacked IF signals is summed without regard to phase and then detected. This pre-detection method has the disadvantage of requiring distinct electronics to convert from RF to IF for each antenna, but avoids the need for estimation of carrier phase and amplitude as well as avoids estimating the signal quality. Here the signals from each receiving element are translated to distinct IF bands, the centers of which must be separated in frequency by multiples of the bit rate. The I.F. signals are combined and then detected by a differential detector. This method is described in the paper by T. Masamura, xe2x80x9cFrequency Offset Receiver Diversity for Differential MSKxe2x80x9d, IEEE Trans. Veh. Technol., Vol. VT-36, No. 2, May 1987, pp. 63-70. This technique is referred to as a frequency stacked IF combiner.
The present invention encompasses a method and preferred embodiment that provides for combining spatial diversity signals.
In accordance with the present invention, diversity signals from receiving elements are combined prior to detection obviating the need for carrier signal recovery or detection prior to modulation signal detection.
In another aspect of the present invention, spatially diverse received signals are directly summed or combined without the need to estimate the phase of the signals from each receiving element, thus eliminating the requirements of the prior art apparatus and methods to obtain phase information and perform phase adjustments prior to combining.
It is yet another aspect of the present invention that spatially diverse received signals are summed or combined without the need to determine or estimate received signal quality of any of the signal receiving elements to effect signal combining thereby eliminating the need to estimate or determine signal strength or bit error rate for each or to any of the received signals.
In yet another aspect of the present invention, a receiver can be implemented using a single down converter to an intermediate frequency (IF) stage providing a significant advantage in receiver construction and economy over receivers requiring multiple IF stage receivers.
The receiver of the present invention includes arrangements including multiple diverse receiving elements and is not limited to two receiving elements.
An object of the present invention is to substantially reduce signal fading in a receiver through the economical implementation of spatial diversity receiving elements.
The present invention includes a method and apparatus to carry out the method and has the advantage of simple implementation. The method comprises perturbing the phase of the carrier of a received signal from a diversity receiving element such that the perturbation of the phase of the carrier is periodic with a period equal to, or a multiple of, the symbol rate. For a plurality of diversity receiving elements, each of the periodic phase perturbations is different. Improved performance in accordance with the invention is obtained when the phase perturbations of the diversity receiving elements are mutually orthogonal over a symbol period. The perturbed phase signals are summed and then detected using a suitable detector. The detector can be, for example, a differential detector.
In a preferred embodiment, binary phase perturbations are applied to each received signal by switching the signal between two circuit paths, one of which has more delay due to extra length. The difference in delay between the two paths is one-half of the carrier cycle. The circuit path switches are controlled by a set of binary functions that are mutually orthogonal over one symbol period (for example, over one data bit interval for coding schemes providing a single bit per symbol) and are each periodic with a common period equal to the symbol period. The binary functions are generated from a square clock using standard logic elements. While there are many sets of binary functions that will work, a set of functions known as Walsh functions are advantageously employed to achieve the advantages of the present invention based on binary phase perturbations.
The present invention discloses a method to combine the signals from each antenna prior to down conversion. In the preferred embodiment, the phase perturber and combiner are placed immediately after the antennas so that only one low noise amplifier is required. In an alternate embodiment, the phase perturber and combiner are placed after a set of low noise amplifiers. Placement of the phase perturber and combiner after low noise amplification of each received signal increases receiver cost but decreases the noise figure of the receiver system.
The method of the invention involves perturbing or modulating the phase of the carriers from each antenna in a periodic manner, without regard for the phase relationship among the carriers received at the different antennas. The method will reduce signal fading as long as the phase of the carriers are perturbed differently and the period of the perturbations is substantially the same as the symbol rate or a multiple thereof. The maximum possible reduction in fading is obtained if the phase perturbations of the carriers are orthogonal providing the magnitude of the perturbation is sufficient.
Theory of Operation
It is believed that the invention works according to the following theory. The invention is not to be limited by the theory, but rather resides in the apparatus and a series of steps comprising the method.
In accordance with the theory, a signal received at any given receiving element i in a plurality of receiving elements as function of time may be expressed as:
ri(t)=Ai cos(xcfx890t+xcfx86i)xe2x80x83xe2x80x83(1)
where:
Ai is the amplitude of the carrier received at receiving element i
xcfx890 is the angular frequency of the carrier
xcfx86j is phase in radians of the carrier received at receiving element i. Where the channel linking the receiving element to the transmitter is slowly fading, the carrier amplitude and phase are essentially constant for the duration of two symbols. Thus the phase xcfx86i will be substantially constant over 2 successive symbol periods facilitating differential detection.
When receiver diversity is employed, there are signals from N receiving elements which may be summed to produce a combined signal which is given by:       s    ⁢          (      t      )        =            ∑              i        =        1            N        ⁢          xe2x80x83        ⁢                  A        i            ⁢              cos        ⁢                  (                                                    ω                0                            ⁢              t                        +                          φ              i                                )                    
The energy of a symbol in the combined signal is given by:       E    s    =            ∫      0      T        ⁢                            s          2                ⁡                  (          t          )                    ⁢              xe2x80x83            ⁢              ⅆ        t            ⁢              xe2x80x83            ⁢      where      ⁢              xe2x80x83            ⁢      T      ⁢              xe2x80x83            ⁢      is      ⁢              xe2x80x83            ⁢      the      ⁢              xe2x80x83            ⁢      symbol      ⁢              xe2x80x83            ⁢      period      
Substituting for s(t) and expressing the square of a sum as a double sum, the above integral may be used to express the combined symbol energy of all signals from N receiving elements as follows:       E    s    =            ∫      0      T        ⁢                  ∑                  j          =          1                N            ⁢              xe2x80x83            ⁢                        ∑                      i            =            1                    N                ⁢                  xe2x80x83                ⁢                              A            i                    ⁢                      A            j                    ⁢          cos          ⁢                      xe2x80x83                    ⁢                      (                                                            ω                  0                                ⁢                t                            +                              φ                i                                      )                    ⁢          cos          ⁢                      xe2x80x83                    ⁢                      (                                                            ω                  0                                ⁢                t                            +                              φ                j                                      )                    ⁢                      xe2x80x83                    ⁢                      ⅆ            t                              
Given the trigonometric identity: cos(a)cos(b)=xc2xd[cos(a+b)+cos(axe2x88x92b)], the previous equation can be expressed as:                     E        s            =                        ∫          0          T                ⁢                              ∑                          j              =              1                        N                    ⁢                      xe2x80x83                    ⁢                                    ∑                              i                =                1                            N                        ⁢                          xe2x80x83                        ⁢                                                                                A                    i                                    ⁢                                      A                    j                                                  2                            [                                                cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  ω                          0                                                ⁢                        t                                            +                                              φ                        i                                            +                                              φ                        i                                                              )                                                  +                                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  φ                        i                                            -                                              φ                        j                                                              )                                                                                            ]    ⁢      xe2x80x83    ⁢      ⅆ    t  
For communications signals, the carrier frequency is selected to be much, much greater than the symbol rate. This may be expressed mathematically as: xcfx890 greater than  greater than 2xcfx80/T. Thus, over a symbol period, integration of the term involving cos(2xcfx890t . . . ) will go to zero. Changing the order of summation and integration of the foregoing formula and recognizing that for i=j, cos(xcfx86ixe2x88x92xcfx86j)=cos(0)=1, allows Es to be expressed as:                               E          s                =                                            ∑                              i                =                1                            N                        ⁢                          xe2x80x83                        ⁢                                          ∫                0                T                            ⁢                                                                    A                    i                    2                                    ⁢                                      ⅆ                    t                                                  2                                              +                                    ∑                              j                =                1                            N                        ⁢                          xe2x80x83                        ⁢                                          ∑                                                      i                    =                    1                                                        i                    ≠                    j                                                  N                            ⁢                              xe2x80x83                            ⁢                                                ∫                  0                  T                                ⁢                                  xe2x80x83                                ⁢                                                                                                    A                        i                                            ⁢                                              A                        j                                                              2                                    ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  φ                        i                                            -                                              φ                        j                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                    t                                                                                                          (        2        )            
It is noted that there are two summands provided in this equation. The summand for the single sum is independent of carrier phase and is always positive irrespective of the receiving element i receiving the signal. The summand for the single sum increases with increasing N. Thus, the received signal strength will increase as N increases. While each Ai will vary with time, the probability that all A will be small drasically goes down as N increases.
On the other hand, the summand for the double sum is a function of the received signal carrier phases and could be negative for one or more combinations of i and j. The value of each term in the double sum depends on the phase difference between the respective carrier phases xcfx86i and xcfx86j. Accordingly, the carrier phases could, and at times would, be such that the double sum could contain a sufficient number of negative terms which provide a sum that completely counteracts the single sum thereby forcing Es to zero, or some value near zero. When Es approaches zero, by definition, signal fading or signal loss results. Therefore, constraining the terms of the double sum to provide a total which approaches zero, will operate to counteract the negative effect the double sum has on the single sum thereby reducing or eliminating fading and signal loss.
Phase modulation apparatus can be provided to modify or adjust the phase of the received signal. This can be done by introducing a time varying delay to the received signal. The received signal represented by equation (1) can be rewritten to include the time varying phase function as follows:
ri(t)=Aj cos(xcfx890t+xcfx86j+xcexa8i(t))
where:
xcexa8i(t) expresses the instantaneous phase adjustment for the signal received at each receiving element i as a function of time where a time varying phase adjustment is introduced.
When the received signal for each receiving element i is individually phase adjusted, it can be shown that the combined symbol energy of all signals from N receiving elements, as previously given in equation (2), can then be represented by:                               E          s                =                  "AutoLeftMatch"                                                    ∑                                  i                  =                  1                                N                            ⁢                              xe2x80x83                            ⁢                                                ∫                  0                  T                                ⁢                                                                            A                      i                      2                                        ⁢                                          ⅆ                      t                                                        2                                                      +                                          ∑                                  j                  =                  1                                N                            ⁢                              xe2x80x83                            ⁢                                                ∑                                                            i                      =                      1                                                              i                      ≠                      j                                                        N                                ⁢                                  xe2x80x83                                ⁢                                                      ∫                    0                    T                                    ⁢                                      xe2x80x83                                    ⁢                                                                                                              A                          i                                                ⁢                                                  A                          j                                                                    2                                        ⁢                                          xe2x80x83                                        ⁢                    cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        φ                          i                                                -                                                  φ                          j                                                +                                                                              Ψ                            i                                                    ⁡                                                      (                            t                            )                                                                          -                                                                              Ψ                            j                                                    ⁡                                                      (                            t                            )                                                                                              )                                        ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      t                                                                                                                              (        3        )            
By examination of equation (3), it is observed that by introducing individual phase adjustment of the received signal from each receiving element i, the summand for the double sum becomes a function of both the individual received signal carrier phases, xcfx86i and xcfx86j, and the individual signal phase adjustment functions, xcexa8i(t) and xcexa8j(t).
The invention comprises utilizing a family of functions, preferably Walsh functions, that may be employed advantageously to provide individual signal phase adjustment which operates to constrain the double sum to approach zero for all permutations of the signal carrier phases xcfx86ixe2x88x92xcfx86j. Use of this family of functions operates to minimize the counteracting effect the double sum has on the single sum thereby reducing or eliminating fading and signal loss.
In accordance with the invention, a family of mutually orthogonal periodic functions has the following properties:
1. A first function has a single output state which is constant, either: high or low, one or zero etc. This can be expressed as follows:       f    0    =      {                                        1            ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              100              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                        ;            or                                                            0            ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              100              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                                          
2. For any other order function (f1, f2, f3 . . . fN), the output state will be high for one-half the time and low for one-half the time. This may be expressed mathematically as follows:       f    i    =      {                                        1            ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              50              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                        ;            and                                                            0            ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              50              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                                          
3. Between any two higher order functions, over a given time period, the output states of each individual function will be equal one-half the time and opposite one-half the time. This may be expressed mathematically as follows:                     f        i            :              f        j              =      {                                                      0:0                        ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              25              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                        ;                                                                          0:1                        ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              25              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                        ;                                                                          1:0                        ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              25              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                        ;            and                                                                          1:1                        ;                          xe2x80x83                        ⁢                          for              ⁢                              xe2x80x83                            ⁢              25              ⁢              %              ⁢                              xe2x80x83                            ⁢              of              ⁢                              xe2x80x83                            ⁢              the              ⁢                              xe2x80x83                            ⁢              time                                          
The invention provides that such functions be used to produce time varying phase perturbations of the received signal for each receiving element i. The resulting time varying phase perturbation may be expressed as: xcexa8i(t). The differences in the perturbated phases between any two received signals can then be expressed as: xcexa8i(t)xe2x88x92xcexa8j(t). These functions are selected to be periodic over a receiver""s symbol period. When such functions are so employed, the individual phase perturbations of each received signal can be expressed as follows:
1. Phase perturbation of a received signal i controlled by function f0 may be expressed as a constant as follows:
xcexa8i(t)=xcexa8i for 100% of a symbol period, where xcexa8i could be 0xe2x80x83xe2x80x83(P1)
2. Phase perturbation of a received signal i controlled by any function, other than f0, may be expressed as:                                           Ψ            i                    ⁡                      (            t            )                          =                  {                                                                      0                  ;                                      xe2x80x83                                    ⁢                                      for                    ⁢                                          xe2x80x83                                        ⁢                    50                    ⁢                    %                    ⁢                                          xe2x80x83                                        ⁢                    of                    ⁢                                          xe2x80x83                                        ⁢                    the                    ⁢                                          xe2x80x83                                        ⁢                    time                                    ;                  and                                                                                                                          Ψ                    i                                    ;                                      xe2x80x83                                    ⁢                                      for                    ⁢                                          xe2x80x83                                        ⁢                    50                    ⁢                    %                    ⁢                                          xe2x80x83                                        ⁢                    of                    ⁢                                          xe2x80x83                                        ⁢                    the                    ⁢                                          xe2x80x83                                        ⁢                    time                                                                                                          (        P2        )            
In accordance with the condition specified in P2, the phase perturbation of any individual signal occurs for one-half of the time, or, stated another way, is subject to a fifty percent duty cycle. For the other one-half of the time, no phase adjustment of the signal occurs.
3. Phase perturbation phase differences between a received signal i having a phase perturbation controlled by f0 and any other received signal j having a phase perturbation controlled by a function which is not f0, may be expressed as:                                                         Ψ              i                        ⁡                          (              t              )                                -                                    Ψ              j                        ⁡                          (              t              )                                      =                  {                                                                                          Ψ                    i                                    ;                                                                              for                  ⁢                                      xe2x80x83                                    ⁢                  50                  ⁢                  %                  ⁢                                      xe2x80x83                                    ⁢                  of                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  symbol                  ⁢                                      xe2x80x83                                    ⁢                  period                                                                                                                                                Ψ                      i                                        -                                          Ψ                      j                                                        ;                                                                              for                  ⁢                                      xe2x80x83                                    ⁢                  50                  ⁢                  %                  ⁢                                      xe2x80x83                                    ⁢                  of                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  symbol                  ⁢                                      xe2x80x83                                    ⁢                  period                                                                                        (P3)            
4. Phase perturbation phase differences between any two signals i and j each having phase perturbations controlled by different functions, neither of which is f0, may be expressed as:                                                         Ψ              i                        ⁡                          (              t              )                                -                                    Ψ              j                        ⁡                          (              t              )                                      =                  {                                                                      0                  ;                                                                              for                  ⁢                                      xe2x80x83                                    ⁢                  25                  ⁢                  %                  ⁢                                      xe2x80x83                                    ⁢                  of                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  symbol                  ⁢                                      xe2x80x83                                    ⁢                  period                  ⁢                                      xe2x80x83                                    ⁢                                      (                    a                    )                                                                                                                                            Ψ                    i                                    ;                                                                              for                  ⁢                                      xe2x80x83                                    ⁢                  25                  ⁢                  %                  ⁢                                      xe2x80x83                                    ⁢                  of                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  symbol                  ⁢                                      xe2x80x83                                    ⁢                  period                  ⁢                                      xe2x80x83                                    ⁢                                      (                    b                    )                                                                                                                                            -                                          Ψ                      j                                                        ;                                                                              for                  ⁢                                      xe2x80x83                                    ⁢                  25                  ⁢                  %                  ⁢                                      xe2x80x83                                    ⁢                  of                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  symbol                  ⁢                                      xe2x80x83                                    ⁢                  period                  ⁢                                      xe2x80x83                                    ⁢                                      (                    c                    )                                                                                                                                                                  Ψ                      i                                        -                                          Ψ                      j                                                        ;                                                                              for                  ⁢                                      xe2x80x83                                    ⁢                  25                  ⁢                  %                  ⁢                                      xe2x80x83                                    ⁢                  of                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  symbol                  ⁢                                      xe2x80x83                                    ⁢                  period                  ⁢                                      xe2x80x83                                    ⁢                                      (                    d                    )                                                                                                          (P4)            
In accordance with the condition specified in P4, the phase perturbation between any two phase perturbed signals has 4 possible outcomes each of which occurs equally, exactly 25% of the time in one symbol period. Condition P2 still applies to each individual signal and requires each to be phase perturbed for one half of the time. In other words, subject to a fifty percent duty cycle. Condition P4 requires that the phase perturbation between any two signals be mutually absent for twenty-five percent of the time, mutually present for twenty-five percent of the time, and present in one but not the other for twenty-five percent of the time and present in the other but not the one for twenty-five percent of the time. For the purposes of the invention herein described, any two binary functions are said to be mutually orthogonal when they meet the requirements of condition P4.
Without loss of generality, it can be assumed that function xcexa81(t), which represents the phase adjustment of a first received signal (i.e. i=1), is a constant corresponding to function f0. Accordingly, xcexa81(t)=xcexa81. This meets the requirements of the condition expressed in equation P1. Further, and again without loss of generality, it can be assumed that xcexa82(t) through xcexa8N(t), which represent the phase adjustment of the second through N received signals, is given by the functions f1, f2, . . . , fNxe2x88x921. Accordingly, xcexa8i(t)=xcexa8i for 50% of a symbol time and xcexa8i(t)=0 for 50% of a symbol time. This meets the requirements of the condition expressed in equation P2.
Based on the simplifying assumptions of the previous paragraph, and incorporating the individual received signal phase perturbations defined by equations P3 and P4, the double sum summand expressed in equation (3) can be expressed as:                               E          φ                =                              2            ⁢                                          ∑                                  j                  =                  2                                N                            ⁢                                                ∫                  0                                      T                    2                                                  ⁢                                                                                                                              A                          1                                                ⁢                                                  A                          j                                                                    2                                        ⁡                                          [                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                1                                                            -                                                              φ                                j                                                            +                                                              Ψ                                1                                                                                      )                                                                          +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                1                                                            -                                                              φ                                j                                                            +                                                              Ψ                                1                                                            -                                                              Ψ                                j                                                                                      )                                                                                              ]                                                        ⁢                                      ⅆ                    t                                                                                +                                    ∑                              j                =                2                            N                        ⁢                                                            ∑                                      i                    =                    2                                    N                                                  i                  ≠                  j                                            ⁢                                                                                          A                      i                                        ⁢                                          A                      j                                                        2                                ⁢                                                      ∫                    0                                          T                      4                                                        ⁢                                                            [                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                                                      )                                                                          +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                            +                                                              Ψ                                i                                                                                      )                                                                          +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                            -                                                              Ψ                                j                                                                                      )                                                                          +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                            +                                                              Ψ                                i                                                            -                                                              Ψ                                j                                                                                      )                                                                                              ]                                        ⁢                                          ⅆ                      t                                                                                                                              (        4        )            
where Excfx86 is used as a symbol for the phase dependent double sum term, being the second summand or double sum of equation (3)
Given the trigonometric identity:
cos(a+xcfx80)=cos(axe2x88x92xcfx80)=xe2x88x92cos(a),xe2x80x83xe2x80x83(5)
it will be understood that the carrier phase dependent term, Excfx86 of equation (4), is forced to zero for any value of xcexa81 if:
xcexa8i=xcfx80 for i=2, . . . , N
For clarity, substitution of xcfx80 for xcexa8i in equation (4) results in:       E    φ    =            2      ⁢                        ∑                      j            =            2                    N                ⁢                              ∫            0                          T              2                                ⁢                                                                                          A                    1                                    ⁢                                      A                    j                                                  2                            ⁡                              [                                                      cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        φ                          1                                                -                                                  φ                          j                                                +                                                  Ψ                          1                                                                    )                                                        +                                      cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        φ                          1                                                -                                                  φ                          j                                                +                                                  Ψ                          1                                                -                        π                                            )                                                                      ]                                      ⁢                          ⅆ              t                                            +                  ∑                  j          =          2                N            ⁢                                    ∑                          i              =              2                        N                                i            ≠            j                          ⁢                                                            A                i                            ⁢                              A                j                                      2                    ⁢                                    ∫              0                              T                4                                      ⁢                                          [                                                      cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        φ                          i                                                -                                                  φ                          j                                                                    )                                                        +                                      cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        φ                          i                                                -                                                  φ                          j                                                +                        π                                            )                                                        +                                      cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        φ                          i                                                -                                                  φ                          j                                                -                        π                                            )                                                        +                                      cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        φ                          i                                                -                                                  φ                          j                                                +                        π                        -                        π                                            )                                                                      ]                            ⁢                              ⅆ                t                                                        
Performing the substitutions of the trigonometric identity given in equation (5) results in:                               E          φ                =                              2            ⁢                                          ∑                                  j                  =                  2                                N                            ⁢                                                ∫                  0                                      T                    2                                                  ⁢                                                                                                                              A                          1                                                ⁢                                                  A                          j                                                                    2                                        ⁡                                          [                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                1                                                            -                                                              φ                                j                                                            +                                                              Ψ                                1                                                                                      )                                                                          +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                1                                                            -                                                              φ                                j                                                            +                                                              Ψ                                1                                                                                      )                                                                                              ]                                                        ⁢                                      ⅆ                    t                                                                                +                                    ∑                              j                =                2                            N                        ⁢                                                            ∑                                      i                    =                    2                                    N                                                  i                  ≠                  j                                            ⁢                                                                                          A                      i                                        ⁢                                          A                      j                                                        2                                ⁢                                                      ∫                    0                                          T                      4                                                        ⁢                                                            [                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                                                      )                                                                          -                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                                                      )                                                                          -                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                                                      )                                                                          +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          φ                                i                                                            -                                                              φ                                j                                                                                      )                                                                                              ]                                        ⁢                                          ⅆ                      t                                                                                                                              (        6        )            
By inspection of either equation (4) or (6), it is further understood that Excfx86 is forced to zero for any values of xcexa81. Thus, with xcexa8i=xcfx80 for i=2, 3, . . . N, the cosine terms in the integrands sum to zero. This makes the integrals for all i, j equal to zero and thereby forces Excfx86 to zero. This is the desired objective.
A preferred embodiment of the present invention will now be described with reference to the drawings in which like features of the invention bear like reference numerals throughout the various figures.